Ground state solutions for a class of Schrödinger-Poisson systems with Hartree-type nonlinearity

Abstract

In this paper, we consider the following Schrödinger-Poisson system with Hartree-type nonlinearity $\begin{array}{ll}-\mathrm{\Delta }u+u+\lambda \phi u=(I_{\alpha }\ast |u{|}^p)|u{|}^{p-2}u,& \text{in}{\mathbb{R}}^3,\\ -\mathrm{\Delta }\phi =u^2,& \text{in}{\mathbb{R}}^3,\end{array}$ where $\lambda >0$, $0<\alpha <3$, $I_{\alpha }$ is a Riesz potential and $\frac{3+\alpha }{3}<p<3+\alpha$. By using the Pohozaev type identity and the filtration of Nehari manifold, we show the existence of positive ground state solutions for the above system.

COMMUNICATED BY:

• Daomin Cao

Keywords:

• Ground state
• Schrödinger-Poisson equations
• Pohozaev type identity
• Nehari manifold
• Hartree-type

2010 Mathematics Subject Classifications:

• 35J60
• 35Q40
• 35J20

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Weihong Xie  http://orcid.org/0000-0002-1762-988X

Tsung-Fang Wu  http://orcid.org/0000-0003-4945-652X

Additional information

Funding

H.B. Chen is supported by the National Natural Science Foundation of China (No. 11671403); T.F. Wu was supported in part by the Ministry of Science and Technology, Taiwan (No. 106-2115-M-390-001-MY2) and the National Center for Theoretical Sciences, Taiwan.